On 13 May 2004, at 23:26, Daniel Lichtblau wrote:
>
> The issue of 1`2==2.2 is a bit dicey. I believe the most useful
> interpretation is roughly as Andrzej suggests, that the model breaks
> down at sufficiently low precision. Yes, there are "contradictions" is
> the logic that lead to the equality above, but to me they are not
> terribly compelling.
>
>
> Daniel
>
>
At least in this respect Mathemaitca's behaviour agrees with the
documentation: since
RealDigits[1`2,2]
{{1,0,0,0,0,0,0},1}
has only 7 digits it does "differ in at most last eight binary digits"
from 2.
This is a bit more tricky
1.`2 == 0
True
Sin[1`2]==0
False
The key point is clearly the fact that here Sin slightly increases
precision:
Precision[Sin[1`2]]
2.1924
Looking at RealDigits[Sin[1`2],2] is not quite convincing, since
Mathematica still returns only 7 digits:
RealDigits[Sin[1`2],2]
{{1,1,0,1,1,0,0},0}
but I suspect that the statement in the documentation about "last eight
binary digits" is also only approximate and that this is actually a
border line case (just over the border line).
In any case, these are clearly pathologies of significance arithmetic
and without importance.
Andrzej Kozlowski
Chiba, Japan
http://www.mimuw.edu.pl/~akoz/